Since \(\mathrm {coker}(\varphi |_{\eta _{0}})\)  is not admissible, \(\mathrm {coker}(\varphi |_{\eta _{0}})\)  is not normal to \(\mathcal {D}_{-}\) , or \(\mathcal {D}_{+}\) , or both of them. We will deal with the case where \(\mathrm {coker}(\varphi |_{\eta _{0}})\)  is not normal to \(\mathcal {D}_{-}\) , the remaining cases are the same. As normality has the local property in the \(\acute{e}\) tale topology, we consider an affine \(\acute{e}\) tale neighborhood of a point in \(\mathcal {D}_{-}\times \eta _{0}\) . As in the proof of [[1]}, Lemma 5.16], the local model of \(\Delta \times S\)  can be chosen to be \(U=\mathrm {Spec}\,A:=\mathrm {Spec}\,R[y,\mathbf {z}]\)  where \((\mathcal {D}_{-}\times \eta _{0})|_{U}\)  is defined by the ideal \((y,u)\) , the vector \(\mathbf {z}\)  represents the coordinates in \(\mathcal {D}\)  (containing \(\mathcal {D}_{-}\) ) and \(y=0\)  is the local equation defining \(\mathcal {D}\) . Suppose the coherent sheaf \(\mathrm {coker}\,\varphi \)  on \(U\)  is represented by \(A\) -module \(M\) . Since \(\varphi :\mathcal {O}_{\Delta \times S}\rightarrow \mathcal {F}\)  is an \(S\) -flat family of orbifold PT pairs and the map \((\mathrm {Im}\,\varphi )|_{s}\rightarrow \mathcal {F}|_{s}\)  restricted on \(U\)  is injective for each \(s\in S\) , then the coherent sheaf \(\mathrm {coker}\,\varphi \)  on \(U\)  is flat over \(S\)  by [[2]}, (20.E)].
That is, the \(A\) -module \(M\)  is flat over \(R\) .
Since \(\dim \mathrm {coker}\,\varphi =0\) ,
by [[3]}, Theorem 2.14 and Corollary 2.17] or [[2]}, (12.B)], the \(A\) -module \(M\)  is of finite length.
Then there exists an integer \(l>0\)  such that we have the surjection \(A^{\oplus l}\rightarrow M\rightarrow 0\) . Denote by \(K\)  the kernel of this surjection, and we have the following short exact sequence
\(0\rightarrow K\rightarrow A^{\oplus l}\rightarrow M\rightarrow 0.\) 
Now, the proof is completed by applying the similar argument in [[1]}, Lemma 5.19] where we obtain the normality of \(\mathrm {coker}\,\widetilde{\varphi }\)  (viewed as some quotient sheaf due to the above short exact sequence) to \(\mathcal {D}_{-}\times \eta _{0}\)  via using the properness of moduli space \(\mathrm {PT}_{\Delta /\mathbb {C}}\)  (see Section 5.1 for the definition) to extend \(\widetilde{\varphi }|_{\eta }\)  (it is still an orbifold PT pair) over \(\eta _{0}\)  and taking the flat limit of \(\mathrm {coker}(\widetilde{\varphi }|_{\eta _{0}})\) , where \(\widetilde{\varphi }|_{\eta }\)  is related to \(\varphi |_{\eta }\)  through some \(\mathbb {C}^*\) -action on \(\Delta \)  (hence on \(\mathcal {X}(1)\rightarrow \mathbb {A}^2\) ).
